Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.

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Show that a subspace is closed in a Hilbert space $H$

If $T$ is a bounded linear operator in a Hilbert space $H$, and $T$ is self-adjoint and is equal to its inverse, how can I show that $\widehat{H} = \{h + Th : h \in H\}$ is closed? If I consider the ...
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17 views

Normal operators on a hilbert space over the reals - does $norm(Tx)=norm(T^*x)$ imply $T$ normal?

The title states the question. It's easy to prove the result for scalars C via polarisation identities but I don't think the same method works in the real case: Let $S=TT^*-T^*T$ then one obtains $&...
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134 views

Prob. 9, Sec. 4.3 in Kreyszig's Functional Analysis Book: Proof of the Hahn Banach Theorem without Zorn's Lemma [duplicate]

Here's Theorem 4.3-2 (i.e. the Hahn Banach theorem for normed spaces): Let $f$ be a bounded linear functional defined on a subspace $Z$ of a normed space $X$. Then there exists a bounbed linear ...
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Prob. 6, Sec. 4.3 in Kreyszig's Functional Analysis Book: What are all possible extensions?

Here's Theorem 4.3-2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space, let $Z$ be a subspace of $X$, and let $f$ be a bounded linear functional ...
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52 views

Domain Issue: Notation

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$A:\mathcal{D}A\subseteq\mathcal{H}\to\mathcal{K}$$ It is well known that:* $$A=A^{**}\iff A\in\mathcal{C}(\...
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17 views

Normal operator and real eigenvalues [duplicate]

If all eigenvalues of normal operator are real, will it imply operator is self adjoint
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29 views

Wave Operators: Unitarity

This thread is Q&A. Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: $$U_\#(...
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57 views

Dense Operators: Kernel

This thread is Q&A. Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$A:\mathcal{D}A\subseteq\mathcal{H}\to\mathcal{K}$$ Then for the kernel: $$\mathcal{N}A=...
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56 views

If $(\lambda_n)_{n=1}^\infty$ is a bounded sequence, then there is a bounded linear operator $A$ on a Hilbert space $H$ such that $Ae_n=\lambda_n e_n$

If $(\lambda_n)_{n=1}^\infty$ is a bounded sequence, then there is a bounded linear operator $A$ on a Hilbert space $H$ such that $Ae_n=\lambda_n e_n$ for all $n\in \mathbb{N}$. Let ${e_n}$ be a ...
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67 views

Boundedness of $A$ in the operator equation $Au = f$ of $-\Delta u(x)=f(x)$.

We consider the boundary value problem on a bounded, open domain $\Omega \subset \mathbb R^n$ determining $u : \Omega \rightarrow \mathbb R$ such that $$-\Delta u(x)=f(x), \qquad u|_{\partial\Omega}=0,...
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115 views

How to prove a set is a core for an infinitesimal generator

I am trying to prove that the set $D=\bigcap_{n\geq 1} \{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f(\frac{k}{\sqrt{n}})g_n (k)\}$ is a core for the infinitesimal generator of reflecting brownian ...
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22 views

Wave Operators: Isometry

This thread is only Q&A. Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}H_\#\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: $$U_\...
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38 views

Isometries: Weak vs. Strong

Given a Hilbert space $\mathcal{H}$. Consider isometries: $$R_\lambda\in\mathcal{B}(\mathcal{H}):\quad R_\lambda^*R_\lambda=1$$ Then it follows: $$R_\lambda\rightharpoonup R\implies R_\lambda\to R$$ ...
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51 views

Is $B(H)$ the weak-$*$ closure of $K(H)$?

I am getting the following result: If $H$ is a Hilbert space, then the weak-$*$ closure of $K(H)$, the space of compact operators on $H$, is $B(H)$, the space of bounded operators on $H$. Is this ...
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43 views

Show by example that $AB=I$ does not imply that $BA=I$, with $I$ being the identity operator on $Y$. What is a suitable $Y$ for this to hold?

Let $A$ and $B$ be bounded linear operators on a normed space $Y$ into $Y$. Show by example that $AB=I$ does not imply that $BA=I$, with $I$ being the identity operator on $Y$ Here is what I have ...
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172 views

Polar Decomposition: Adjoint

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider a closed operator: $$A:\mathcal{D}(A)\subseteq\mathcal{H}\to\mathcal{K}:\quad A=A^{**}$$ Polar decompose: $$A=J|A|:\quad J^*J=...
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83 views

What is the relation between the matrix of a bounded linear operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, both real or both complex, and let $\dim X = n$ and $\dim Y = m$. Let $E \colon= ( e_1, \ldots, e_n )$ be an ordered basis for $X$, and let $F \...
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1answer
39 views

Weyl sequence for closure of an operator

I'm trying to solve following exercise and need some hints. Let $A= \bar{ A_0 }$ be closure of $A_0$ - a densely defined operator. Suppose $f_n \in D(A)$ is Weyl sequence for $z \in \sigma (A)$. Show ...
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42 views

cauchy's integral formula in operator theory

Can you prove cauchy's integral formula based on the assumptions in Conway book, operator theory, VII, 4.2? 4.2. Cauchy's Integral Formula If $\mathcal X$ is a Banach space, $G$ is an open subset ...
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1answer
120 views

Closure of the Hamilton's operator $(Hf)(x)=\frac{1}{2}f''(x)-V(x)f(x)$ with $C_c^\infty(\mathbb{R}, \mathbb{C})$ domain

Let $V \in C_{b}^{1}(\mathbb{R}, \mathbb{R})$ be a differentiable function bounded with its first derivative and $H$ be a Hamilton's operator such that: $$(Hf)(x)=\frac{1}{2}f''(x)-V(x)f(x)$$ The ...
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1answer
31 views

Uniform closure of polynomials

What is the meaning of "uniform closure of polynomials"? I have seen it in Conway's Functional Analysis book VII § 5.
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61 views

Prob. 8, Sec. 4.5 in Kreyszig's functional analysis book: The inverse of the adjoint operator is the adjoint of the inverse operator

Let $X$ and $Y$ be normed spaces, both real or both complex, let $B(X,Y)$ denote the space of all the bounded linear operators $T \colon X \to Y$, and let $T^\times$ denote the adjoint operator of $T$....
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126 views

How to apply Theorem 4.3-3 in the proof of Theorem 4.5-2 in Kreyszig's functional analysis book?

Here's Theorem 4.3-3 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space and let $x_0 \neq 0$ be any element of $X$. Then there exists a bounded ...
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154 views

Is the Laplacian an unbounded operator?

"The Laplacian is an unbounded operator": I read this in a book. But on Wikipedia it says: The Laplace operator $$\Delta:H^2({\mathbb R}^n)\to L^2({\mathbb R}^n) \,$$ (its domain is a Sobolev ...
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1answer
155 views

Self-adjoint operator has non-empty spectrum.

I am trying to prove, that a self-adjoint (maybe unbounded) operator has a non-empty spectrum. So far I have argued, that if $\sigma(T)$ would be empty, $T^{-1}$ would be a bounded self-adjoint ...
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28 views

Operator not bounded below on sum of subspaces

Let $X$ be a Banach space. Say that a bounded linear operator $T\colon X\to X$ is bounded below by $\delta>0$ on $Y\subset X$ if $\|Tx\|\geqslant \delta \|x\|$ for all $x\in Y$. Is there a Banach ...
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36 views

Embedding: Extension

Problem Given Banach spaces $E_0$ and $E$ Regard dense domain: $$\overline{\mathcal{D}_0}=E_0\quad\overline{D}=E$$ Consider an embedding: $$\Phi:\mathcal{D}_0\hookrightarrow\mathcal{D}:\quad\|\Phi(\...
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80 views

Compact linear operators $S$ and $T$, show that $S(I-T)=I$ if and only if $(I-T)S=I$ and deduce that $I-(I-T)^{-1}$ is a compact operator

If $T:X\to X$ is a compact linear operator, then for any bounded linear operator $S:X\to X$ we have that $S(I-T)=I$ if and only if $(I-T)S=I$. Where $X$ is a normed space, also $T$ is bounded. With ...
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1answer
40 views

Generalization of closed range theorem

For Hilbert spaces $X$ and $Y$, the closed range theorem is ok if the operator $T:X\rightarrow Y$ has a closed range. But do we in general have \begin{equation} Y=\overline{\operatorname{ran}(T)}\...
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1answer
50 views

uniform boundedness principle - problem with understanding proof

I'm looking at the proof and there is a step I really don't get. He states that for $\parallel x-x_0\parallel<\epsilon\Rightarrow \parallel T_\alpha x\parallel$, where $T_\alpha$ Is a bounded ...
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1answer
74 views

$‎\Delta ‎^2(.)-‎\frac{‎‎\lambda‎‎}{|x|^4}(.): W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \to W_0^{-2,2}(\Omega) ‎$ is coercive.

I am reading an article and there, author claim that $$‎L(.)=\Delta ‎^2(.)-‎\frac{‎‎\lambda‎‎}{|x|^4}(.): W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \to W_0^{-2,2}(\Omega) ‎$$ is coercive if ‎‎$ ‎0\leq ‎‎...
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Is there a name for linear transformations $T: C(\mathbb{R}) \to \mathbb{R}^n$

Briefly, I wish to study forms that can approximate all such linear transformations $T: C(\mathbb{R}) \to \mathbb{R}^n$. Clearly the Riesz representation theorem cannot be applied in this case as $\...
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2answers
151 views

Example of self-adjoint linear operator with pure point spectrum on an infinite-dimensional separable Hilbert space

Let $\mathcal{H}$ be an infinite-dimensional, complex, separable Hilbert space. Besides the well-known one-dimensional Harmonic oscillator on $\mathcal{H}=(\mathcal{L}^{2}(\mathbb{R})\,;d\mathit{l})$, ...
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3answers
66 views

Spectrum: Boundary

Problem Given a Hilbert space $\mathcal{H}$. Denote for readability: $$\Omega\subseteq\mathbb{C}:\quad\|\Omega\|:=\|\omega\|_{\omega\in\Omega}:=\sup_{\omega\in\Omega}|\omega|$$ Denote for shorthand:...
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70 views

On the square root function of matrices

Let $A, B$ be positive definite matrices and let $P$ be an orthogonal projection. If $A \leq PBP,$ does it follow that $$ A^{1/2} \leq PB^{1/2}P?$$
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68 views

Give an example of a non-self-adjoint operator on a Hilbert space $H$ whose range is $H$ and which is not invertible.

Give an example of a non-self-adjoint operator on a Hilbert space $H$ whose range is $H$ and which is not invertible. I cannot think of an example to save my life. Any solutions/hints are greatly ...
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26 views

How to extend formula for residue to functional calculus of operators

Suppose $\{X_t\}$ is a stochastic process with the covariance operator $\Gamma$ and the first $d$ eigen values are $\lambda_1\geq\lambda_2\geq \ldots \geq\lambda_d$ with eigen vectors $\{e_1,\ldots,...
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Closedness of the range of differential operator first order

The fact that the range of gradient from $H_0^1$ to $L_2$ is closed is well known. In general we can define some kind of weak derivative in the form \begin{equation} Du=\sum_{i,j}a_{ij}\partial_i u_j \...
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Proof that every polilinear map who's domain is $R^{n_1} \times R^{n_2}… \times R^{n_k}$ and co-domain any given real normed space Y is bound.

A Polilinear map\operator is $P:X^1 \times ... \times X^n \to Y$ such that the foolowing applies: $\lambda, \mu \in R$ $$ P( \lambda x_1^1 + \mu x_2 ^1, x^2,...,x^n)= \lambda P(x_1^1,x^2,...x^n)+ \mu ...
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67 views

Properties of trace-class operators

Let $X$ be a separable Hilbert space (real or complex). Let $A\in\mathcal{L}\left(X\right)$, a bounded linear operator on $X$, and suppose $B\in\mathcal{L}\left(X\right)$, which is of trace-class. ...
3
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1answer
86 views

How to prove this is a self-adjoint operator?

I have this operator from $H^1_0$ to $H^1_0$ defined by: $$Au(t)=\int_0^1 G(t,s) f(s,u(s))\mathsf ds$$ where $$G(t,s)=\begin{cases} t(1-s), &t\leq s\\s(1-t), &s\leq t.\end{cases}$$ I want to ...
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1answer
61 views

If a sequence of self-adjoint linear operators is convergent, show that its limit is self-adjoint.

If a sequence of self-adjoint linear operators is convergent, show that its limit is self-adjoint. I'm lost on this problem. I don't know how to even start this. Any solutions or hints would be ...
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111 views

non compact closed range operator

Lately I've been reading Abramovich and Aliprantis' book 'An invitation to operator theory', chapter 2 (page 69) on bounded below operators. I would like to find an example of non-compact (and ...
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1answer
63 views

On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded, and find its adjoint. [duplicate]

On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded I know that $||T||\leq 1$, but I don't know how to show this. Any solutions or ...
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1answer
24 views

Wave Operators: Cook

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: $$U_\#(t)^*=U_\#(-t)=U_\#(t)^{-1}...
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1answer
73 views

Ordering: Compactness

Given a Hilbert space $\mathcal{H}$. Denote selfadjoints: $$\mathcal{S}(\mathcal{H}):=\{A\in\mathcal{B}(\mathcal{H}):A=A^*\}$$ Introduce an order: $$A\leq A':\iff\sigma(A'-A)\geq0$$ Denote compacts:...
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1answer
62 views

Singular Spectrum: Criterion

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And its spectral measure: $$E:\mathcal{B}(\mathbb{R})\to\mathcal{B}(\mathcal{H}):\quad H=\...
4
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1answer
103 views

Normal Operators: Meet

Given a Hilbert space $\mathcal{H}$. Normal Operators: $$\mathcal{N}(\mathcal{H}):=\{N:N^*N=NN^*\}$$ Borel Calculus: $$\mathcal{B}(N):=\{\eta({N}):\eta\in\mathcal{B}(\mathbb{C},\mathbb{C})\}$$ ...
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1answer
56 views

Is my proof of closedness of multiplication operator corect?

I am considering an operator $A: L^2(\mathbb R , d \mu) \supset D(A)\to L^2 (\mathbb R, d\mu)$ defined by $(Af)(x)=a(x)f(x)$ for known measurable function $a$. Domain is of course all those functions ...
0
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1answer
55 views

prove the continuity of $T_\phi f=\int_0^1 f(x)\phi(x) \,dx\\$ [duplicate]

Let $\phi\in C[0,1]$ and let $T_\phi~:C[0,1]\rightarrow\mathbb{R}$, defined as $T_\phi f=\int_0^1 f(x)\phi(x) \,dx\\$. How can i prove that it's a continuous operator?